MEN OP MATHEMATICS whose only positive divisors are 1 and the integer itself. " we try to extend this definition to algebraic integers we soon see that we have not found the root of the matter, and we must seek some property of rational primes which can be carried ova: to algebraic integers. This property is the following: if a rational prime p divides the product a x b of two rational integers, then (it can be proved that) p divides at least one of the factors a, 6 of the product. Considering the unit, 1, of rational arithmetic, we notice that 1 has the peculiar property that it divides every rational integer; — 1 also has the same property, and 1, — 1 are the only rational integers having this property. These and other clues suggest something simple that will work, and we lay down the following definitions as the basis for a theory of arithmetical divisibility for algebraic integers. We shall suppose that all the integers considered lie in an algebraic number field of degree n. If T9s:t are algebraic integers such that r = s x t, each of s, t is called a divisor of r, If j is an algebraic integer which divides every algebraic integer La the field, j is called a unit (in that field). A given field may contain an infinity of units, in distinction to the pair 1, — 1 for the rational field, and this is one of the things that breeds difficulties. The next introduces a radical and disturbing distinction between rational integers and algebraic integers of degree greater than 1. An algebraic integer other than a unit whose only divisors are units and the integer itself, is called irreducible. An irreducible algebraic integer which has the property that if it divides the product of two algebraic integers, then it divides at least one of the factors, is called a prime algebraic integer. Ail primes are .irreducibles, but not all irreducibles are primes in some alge- braic number fields, for example in F[V — 5], as will be seen in a moment. In the common arithmetic of 1,2,3 ... the irreducibles and the primes are the same. In the chapter on Fermat the fundamental theorem of (rational) arithmetic was mentioned: a rational integer is the 520